Question

rezolvati ecuatiile
a)x+2x+3x+....+100x=-5050
b)-2x-4x-6x-....-20x=220

Answer 1

a)
x + 2x + .... + 100x = -5050 →
x ( 1 + 2 + ... + 100) = -5050 →
x (100*101)/2 = -5050 →
x * 50 * 101 = -5050 →
x * 5050 = -5050 → x = -1.

b)
-2x - 4x - ... - 20x = 220 →
-2x( 1 + 2 + ... + 10 ) = 220 →
-x * (10*11)/2 = 110 →
-x * 5*11 = 110 →
-x * 55 = 110→
-x = 2 →
x = -2.

Answer 2

Poti folosi suma Gauss, care spune ca:

1 + 2 + 3 + 4 + ... + n = [n · (n + 1)] : 2

a) x+2x+...+100x = -5050
x(100 · 101) / 2 = -5050
10100x = -10100
x = -10100 : 10100
x = -1

b) -2x-4x-6x-....-20x = 220

-(2x + 4x + ... + 20x) = 220

Impartim relatia la 2:

- (x+2x+...+10x) = 110
- x(10 · 11) / 2 = 110
- 110x = 220
x = - 220 : 110
x = -2